zs

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

受欢迎的微积分 >

integral of (3x+1)/(x^2-10x+28)

解答

\int \frac{3 x + 1}{x ^{2} - 10 x + 28} d x

解答

\frac{3}{2} ln x^{2} - 10 x + 28 + \frac{16}{3} arctan (\frac{1}{3} (x - 5)) + C

求解步骤

\int \frac{3 x + 1}{x ^{2} - 10 x + 28} d x

对

x^{2} - 10 x + 28

配方:

(x - 5)^{2} + 3

= \int \frac{3 x + 1}{( x - 5 ) ^{2} + 3} d x

使用换元积分法

= \int \frac{3 u + 16}{u ^{2} + 3} d u

乘开

\frac{3 u + 16}{u ^{2} + 3} : \frac{3 u}{u ^{2} + 3} + \frac{16}{u ^{2} + 3}

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{3 u}{u ^{2} + 3} d u + \int \frac{16}{u ^{2} + 3} d u

\int \frac{3 u}{u ^{2} + 3} d u = \frac{3}{2} ln u^{2} + 3

\int \frac{16}{u ^{2} + 3} d u = \frac{16}{3} arctan (\frac{u}{3})

= \frac{3}{2} ln u^{2} + 3 + \frac{16}{3} arctan (\frac{u}{3})

u = x - 5

代回

= \frac{3}{2} ln (x - 5)^{2} + 3 + \frac{16}{3} arctan (\frac{x - 5}{3})

化简

\frac{3}{2} ln (x - 5)^{2} + 3 + \frac{16}{3} arctan (\frac{x - 5}{3}) : \frac{3}{2} ln x^{2} - 10 x + 28 + \frac{16}{3} arctan (\frac{1}{3} (x - 5))

= \frac{3}{2} ln x^{2} - 10 x + 28 + \frac{16}{3} arctan (\frac{1}{3} (x - 5))

解答补常数

= \frac{3}{2} ln x^{2} - 10 x + 28 + \frac{16}{3} arctan (\frac{1}{3} (x - 5)) + C

作图

流行的例子

derivative 200(e)^{0.03x}

d er i v a t i v e 200 (e)^{0.03 x}

面积 y=x^4-3x^2+3,y=x^2

a re a y = x^{4} - 3 x^{2} + 3, y = x^{2}

d/(dv)((\sqrt[3]{v}-2ve^v)/v)

\frac{d}{d v} (\frac{3 v - 2 v e ^{v}}{v})

tangent y=5x^2-x^3,(1,4)

t an g e n t y = 5 x^{2} - x^{3}, (1, 4)

limit as x approaches 2 of 2/x

x \to 2 lim (\frac{2}{x})